Improved approximations for buy-at-bul and shallow-light $$$$-Steiner trees and $$(,2)$$(,2)-subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light \(k\)-Steiner tree problem (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). The goal is to find a minimum total cost \(r\)-rooted Steiner tree \(H\) containing at least \(k\) terminals, where the cost of each edge \(e\) is \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost \(2\)-edge-connected subgraph with at least \(k\) vertices, which is introduced as the \((k,2)\)-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).